Angle Converter
Type an angle once — see it in degrees, radians, gradians and turns instantly, with exact π fractions for common angles.
Unit circle: common angles
| Degrees | Radians (exact) | Gradians | Turns |
|---|
Four ways to measure the same angle
An angle is a single geometric quantity, but four different unit systems describe it, each built for a different job. Degrees split a circle into 360 parts and are what most people learn in school and use in everyday life. Radians are the "natural" unit in mathematics — a full circle is 2π radians — and are what calculus, trigonometric derivatives, and virtually every programming language's Math.sin()/Math.cos() expect. Gradians (also called gons) split a circle into 400 parts so a right angle is a tidy 100 grad, a scheme designed for decimal-friendly surveying. Turns (or revolutions) simply count full rotations — 1 turn = 360° — and show up in navigation, motor RPM math, and animation timing. This converter keeps all four in sync as you type, plus a degrees-minutes-seconds (DMS) mode for the sexagesimal format used in navigation, astronomy and geographic coordinates.
The conversion formulas
| From | To | Formula |
|---|---|---|
| Degrees | Radians | rad = deg × π / 180 |
| Radians | Degrees | deg = rad × 180 / π |
| Degrees | Gradians | grad = deg × 10 / 9 |
| Degrees | Turns | turn = deg / 360 |
| Degrees | DMS | D = ⌊deg⌋, M = ⌊(deg−D)×60⌋, S = ((deg−D)×60−M)×60 |
Worked examples
Why the exact π fraction only shows up sometimes
Degree-to-radian conversion is always deg × π/180, so mathematically every rational degree value has some exact multiple of π. The catch is that for an arbitrary value like 12.7°, that exact fraction would be 127π/1800 — technically correct but useless to read. This tool only displays the fraction when its reduced denominator is 360 or smaller, which captures the angles people actually search for and memorize (multiples of 15°, plus finer ones like 22.5° = π/8) without cluttering the display with meaningless fractions for everything else.
Common mistakes
- Forgetting to switch calculator mode. Scientific calculators and spreadsheet functions like
SIN()often default to radians; feeding them a raw degree value silently gives the wrong trig result. Always confirm the mode, or convert explicitly first. - Mixing up gradians and grads-per-second units. "Gradian" (angle) and "gradient" (slope) are unrelated despite the similar spelling — this tool only handles the angular gradian (gon).
- Truncating instead of rounding seconds in DMS. 47° 29′ 60″ should carry over to 47° 30′ 0″; this converter normalizes overflowed minutes and seconds automatically so the output never shows an invalid "60″" or "60′".
What this tool doesn't do
This is a unit converter, not a trigonometry solver — it converts a given angle between representations but doesn't compute an angle from two sides of a triangle or from a slope. For angles outside the typical −360°…360° range (multi-turn rotations in animation or robotics, for instance), the degrees, radians, gradians and turns fields still convert correctly; only the unit-circle reference table below is limited to the standard 0°–360° set.
Sources & further reading
- NIST — SI units, including the radian as the unit of plane angle
- ISO 80000-3 — international standard for quantities and units of space and time, including angle
- Khan Academy — trigonometry course covering degree and radian measure
- NOAA National Geodetic Survey — degrees-minutes-seconds coordinates in surveying and geodesy
Frequently asked questions
How do I convert degrees to radians?
Multiply the degree value by π/180 (about 0.0174533). For example 30° × π/180 = π/6 ≈ 0.5236 rad, and 90° × π/180 = π/2 ≈ 1.5708 rad. To go the other way, multiply radians by 180/π. This converter does the multiplication for you and also shows the exact π fraction — π/6, π/4, π/3 and so on — whenever the angle is a common one, instead of just a rounded decimal.
What is a gradian and how does it relate to degrees?
A gradian (also called a gon) divides a full circle into 400 equal parts instead of 360, so a right angle is exactly 100 gradians. The conversion is straightforward: 1 gradian = 0.9 degrees, and 1 degree = 10/9 ≈ 1.1111 gradians. Gradians were designed for decimal-friendly surveying and are still used in some civil engineering and land-surveying software, even though degrees and radians are far more common elsewhere.
How do degrees, minutes and seconds (DMS) work?
A degree splits into 60 minutes of arc, and each minute splits into 60 seconds of arc — the same base-60 structure as clock time. So 47.5° equals 47° 30′ 0″, because 0.5° × 60 = 30′. Switch on DMS mode in this converter to enter or read an angle as degrees, minutes and seconds directly; it stays perfectly in sync with the decimal-degree, radian, gradian and turn fields.
Why do some angles show an exact π fraction and others don't?
Converting degrees to radians is always deg × π/180, so in principle every whole or simple decimal degree value has an exact fraction of π. This tool shows that exact fraction — like π/6 for 30° or 3π/4 for 135° — only when the fraction has a small, tidy denominator (up to 360). For arbitrary decimals such as 12.7° the exact fraction would have an ugly, meaningless denominator, so the tool shows only the decimal radian value instead.
Is my angle value sent to a server?
No. This angle converter runs entirely in your browser — nothing you type is uploaded anywhere. The only thing saved is your last angle and whether DMS mode was on, kept in your browser's local storage so the tool is ready next time you open it.